Analytic continuation of the rotating black hole state counting

Jibril Ben Achour,Karim Noui,Alejandro Perez

doi:10.1007/jhep08(2016)149

Abstract

In loop quantum gravity, a spherical black hole can be described in terms of a Chern-Simons theory on a punctured 2-sphere. The sphere represents the horizon. The punctures are the edges of spin-networks in the bulk which cross the horizon and carry quanta of area. One can generalize this construction and model a rotating black hole by adding an extra puncture colored with the angular momentum J in the 2-sphere. We compute the entropy of rotating black holes in this model and study its semi-classical limit. After performing an analytic continuation which sends the Barbero-Immirzi parameter to γ = ±i,weshowthattheleadingorderterminthesemi-classicalexpansionoftheentropy reproduces the Bekenstein-Hawking law independently of the value of J.

Highlights

Loop Quantum Gravity (LQG) provides a microscopic explanation of the entropy of black holes
When we perform the analytic continuation to γ = ±i following the techniques developed in [35], we show that the leading order term of the black hole entropy reproduces exactly the semi-classical law whatever the value of J is
We define the rules for the analytic continuation, we compute the number of microstates and we study its semi-classical limit using the saddle point approximation

Summary

INTRODUCTION

Loop Quantum Gravity (LQG) provides a microscopic explanation of the entropy of black holes. The isolated horizon definition breaks the diffeomorphism symmetry on the horizon in the rotating case, which ,in the quantum model, makes the number of states uncountably large Another attempt to describe microstates and to compute the entropy of a rotating isolated horizon was presented in [31], where a new angular momentum operator was introduced. The state counting for the model [33] (revisited in Section III in this paper) and for the models [21, 22] (which to our knowledge are the only models compatible with the SU (2) gauge symmetry in the bulk) yields an entropy with an explicit dependence on the angular momentum J of the black hole This result is incompatible with the Bekenstein-Hawking area law.

ENTROPY OF A ROTATING BLACK HOLE

Rules for the analytic continuation

Number of microstates

Semi-classical limit and Bekenstein-Hawking law

THE REAL BARBERO-IMMIRZI PARAMETER CASE

CONCLUSION